| 131 | | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S V G 202 3 GEODETSKI VESTNIK | letn. / Vol. 67 | št. / No. 2 | SI | EN ABSTRACT IZVLEČEK KLJUČNE BESEDE KEY WORDS deformation analysis, Munich approach, modified method, X-method, L-method, testing of congruence, strain analysis of triangles deformacijska analiza, postopek München, modificirana metoda, metoda X, metoda L, testiranje skladnosti, testiranje preoblikovanja trikotnikov UDK: 528.3:004.02 Klasifikacija prispevka po COBISS.SI: 1.01 Prispelo: 18. 4. 2023 Sprejeto: 15. 5. 2023 DOI: 10.15292/geodetski-vestnik.2023.02.131-164 SCIENTIFIC ARTICLE Received: 18. 4. 2023 Accepted: 15. 5. 2023 Simona Savšek, Tomaž Ambrožič MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH Kljub tehnološkemu napredku v stroki ostaja spremljanje stabilnosti gradbeno-inženirskih objektov ena zahtevnejših nalog v inženirski geodeziji. V članku obravnavamo deformacijsko analizo po postopku München, ki jo je predlagal W. M. Welsch. Metoda obravnava testiranje skladnosti geodetske mreže, testiranje preoblikovanja izbranih trikotnikov in testiranje kinematičnih parametrov v 2D-geodetskih mrežah. Deformacijsko analizo po postopku München lahko izvedemo z metodo X, ki temelji na primerjavi koordinat identičnih točk v geodetski mreži med dvema terminskima izmerama, ki so odvisne od geodetskega datuma, ali z metodo L, s katero se ugotavljajo spremembe vrednosti dolžin in kotov, ki so od geodetskega datuma neodvisne količine. V modificirani metodi predlagamo ugotavljanje skladnosti vseh dolžin in vseh kotov v mreži med dvema terminskima izmerama ter ugotavljanje skladnosti vseh trikotnikov med točkami v mreži, ne le izbranih, kot je običaj pri klasičnem pristopu deformacijske analize po postopku München. Predlagane izboljšave testiramo na primeru znanega šestkotnika in izdelamo analizo uspešnosti odkrivanja stabilnih točk v mreži glede na klasični pristop metode München. Rezultati predlaganega postopka se na obravnavanem primeru niso razlikovali od rezultatov drugih postopkov deformacijske analize. Despite the technical progress monitoring the stability of engineering structures remains one of the most difficult tasks in engineering geodesy. This article presents the deformation analysis according to the Munich approach of W. M. Welsch. The method deals with the testing of the geodetic network's congruence, the affinity of selected triangles, and the testing of other kinematic parameters in 2D geodetic networks. Deformation analysis using the Munich approach can be performed using the X-method, which is based on the comparison of the coordinates of identical points in the geodetic network between two sets of measurements that depend on the geodetic datum, or using the L-method, which determines changes in the values of lengths and angles which are quantities independent of the geodetic datum. In the modified method, we propose to determine the congruence of all lengths and all angles in the network between two sets of measurements and to determine the congruence of all triangles between points in the network, not only selected ones, as it is common in the classical approach of deformation analysis based on the Munich approach. The proposed improvements are tested on the example of a known hexagon, and an analysis is made of the success of detecting stable points in the network using the classical Munich approach. In the present case, the results of the proposed method did not differ from the results of other deformation analysis methods. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 132 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N 1 INTRODUCTION Monitoring the stability of engineering structures or natural objects is the systematic measurement and tracking of changes in the shape and/or dimensions of the object in question as a result of loads. Con- tinuous monitoring of deformations and recording the measured values is a prerequisite for deforma- tion analysis, based on which we propose timely maintenance or renovation of the object in question. Deformation monitoring primarily belongs to the field of engineering geodesy, using various measuring devices or sensors. Various methods have been developed for adequate strain analysis (Chrzanowski et al., 1986; Ašanin, 1986; Mihailović and Aleksić, 1994). This paper deals with the deformation analysis according to the Munich approach which was developed by W. Welsch who worked at the Institute of Geodesy at the Military Academy in Munich. The method is based on the analysis of the deformations of triangles whose vertices are formed by geodetic points in the network and whose coordinate changes are calculated based on the angle and length changes in the network. Deformation analysis according to the Munich approach can be performed in two ways (Welsch, 1983; Welsch and Zhang, 1983; Soldo and Ambrožič, 2018): – with the X-method, which is based on the comparison of the coordinates of identical points of the geodetic network of two sets of measurements. Since the coordinates of the points depend on the geodetic datum, the results of the free network adjustment must refer to the same geodetic datum. If the number of points in the network is different for two epochs, the coordinate unknowns of the non-identical points are eliminated using the S-transformation (van Mierlo, 1987), – with the L-method, which is based on comparing quantities independent of the geodetic datum, i.e. bearings and lengths. The classical approach to deformation analysis using the Munich approach requires free network ad- justment for each epoch by providing an identical geodetic datum and testing the homogeneity of the accuracy of the measurements of the dimensions in question. In the following, we perform testing of the congruence of a geodetic network, trying to determine whether displacements and deformations of the whole object have occurred. If we find statistically significant deformations of the object, we proceed to the testing of affinity of the geodesic network by dividing the network into selected triangles and determining the change in the shape of each triangle. We calculate the basic kinematic parameters in each triangle and then other parameters such as normal and shear deformations. Finally, we examine the changes in the datum-independent lengths in the network, as suggested by the author of the W. M. Welsch method, and in this way discover the points that have moved in a statistically significant way (Welsch and Zhang, 1983; Ašanin, 1986). The modified method we propose in this paper is based on forming all possible triangles in the network rather than just the selected ones, and in addition to testing changes in the datum-independent lengths, also testing independent angles in all the triangles formed. In this way, we capture much more infor- mation about the changes in the network and find with greater certainty the points that have moved significantly, which is a considerable improvement on the classical approach to deformation analysis using the Munich approach. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 133 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N According to the two methods of deformation analysis based on the Munich approach, i.e. according to the X-method and according to the L-method, exactly the same results are obtained in the phase of testing of congruence of the geodetic network and the phase of testing of affinity of the geodesic network (Welsch, 1983; Welsch and Zhang, 1983; Soldo and Ambrožič, 2018), therefore, in the proposed modified method, we will use only the X-method for testing of congruence and testing of affinity of the geodetic network. 2 DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH 2.1 Prerequisites for the performance of deformation analyses To perform deformation analysis, it is important to ensure the quality and consistency of measurement accuracy between two sets of measurements. This is done by checking the homogeneity of the accuracy of the measurements with the null hypothesis H0: E(s1 2) = E(s2 2), which states that the accuracy of the measurements between two sets of measurements is the same, where si are a posteriori reference standard deviation of the unit weights for each set of measurements. Gross errors must be excluded from the measurements according to established procedures such as Baard's, Pope's, Danish, or other appropriate methods (Caspary, 1988; Grigillo and Stopar, 2003; Vrce, 2011). Methods of deformation analysis require adjustment of an individual set of measurements as a free network with a minimal trace of the cofactor matrix of unknown (Ambrožič, 2001; Sušić et al., 2017). It is important to consider only the coordinate unknowns, so the orientation unknowns are eliminated by reducing the unknowns in the observation equations. If we consider only length measurements, we must also reduce the unknowns due to the scaling factor in the network (Van Mierlo, 1978). The de- formation analysis can only be performed on identical points in the network, so the S-transformation eliminates coordinate unknowns from non-identical points in the considered set of measurements, if necessary (Van Mierlo, 1978; Caspary 1988; Marjetič and Stopar, 2007; Sušić et al, 2015a; Marković et al, 2019). The results of the adjustment are the estimated vectors of the adjusted coordinates of the points ˆ ix with the corresponding cofactor matrices of the coordinate unknowns x xQ ˆ ˆi i and the a posteriori ref- erence variance of the weight unit si 2 for each set of measurement. After providing an identical geodetic datum and verifying the homogeneity of the accuracy of the two sets of measurements discussed, we compute the reference variance a posteriori using the expression 1 2 2 2 1 1 2 22 1 1 2 2 1 2 , f s f s s f f f + + = = + T T ll llv P v v P v (1) where v1 and v2 are the vectors of residuals of the measurements, Pll1 and Pll2 are the weight matrices of the measurements, f1 and f2 are the number of degrees of freedom and we can write the relation f1 + f2 = f , s1 2 and s2 2 are the a posteriori reference variance after adjustment the 1st and 2nd sets of meas- urements at t1 and t2. If statistically significant deformations are found when testing the conformity of the geodetic network, we continue with the classical approach of testing of affinity of the geodetic network by analysing indi- vidual triangles and determining whether the triangle has significantly changed the shape between two sets of measurements. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 134 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N 2.2 Testing of congruence of the geodetic network The decision of whether displacements and deformations have occurred in the geodetic network is made using the testing of congruence. We formulate the null and alternative hypotheses (Welsch, 1983): x x u 00 1 2ˆ ˆ: ( ) ( ) or ( )H E E E= = ... the coordinates of all points of the network have not changed between two epochs; x x u 01 2ˆ ˆ: ( ) ( ) or ( )aH E E E≠ ≠ ... the coordinates of at least one point of the network have changed between two epochs. When testing the consistency of the X-method with a global stability test for all points in the network, the variance of the point coordinate differences su 2 is compared with a posteriori reference variance estimate s2. Let us compile the test statistics: u u u u Q u2 T 12 1 2 2 , s T s f s − = = ⋅ (2) The vector of displacements of identical points is calculated using the following expression: u x x2 1ˆ ˆ ,= − (3) where x x1 2ˆ ˆand are the vector of the adjusted coordinates of all points after the adjusting 1st and 2nd sets of measurements t1 and t2. The cofactors matrix of the coordinate differences is written as u x x x xQ Q Q1 1 2 2ˆ ˆ ˆ ˆ ,= + (4) where x x x xQ Q1 1 2 2ˆ ˆ ˆ ˆand are the cofactor matrices of the coordinate unknowns after smoothing, which are calculated with the expressions x x ll x x llQ B P B G G Q B P B G G1 1 1 2 2 2 T T 1 T T 1 ˆ ˆ ˆ ˆ1 1 2 2( ) and ( ) − −= + = + . B1 and B2 are the design matrices. Pll1 and Pll2 are the weight matrices of the 1 st and 2nd sets of measurements at moments t1 and t2 (Kuang, 1996). Datum matrix GT in 2D geodetic triangulation network has the form (Welsch, 1986): T 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 i i j j m m i i j j m m x y x y x y y x y x y x ...     ...  =   − − ... −   ...    G , (5) where yi 0, xi 0, yj 0, xj 0, ym 0, xm 0 are the approximate coordinates of m points of geodetic network. The number of degrees of freedom is: fu = u − d. (6) We denote the number of all points in the geodetic network by m. In equation (6) u = 2m represents Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 135 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N the number of unknown coordinates in the 2D network and d the datum defect of the geodetic network (Welsch, 1983; Welsch and Zhang, 1983; Soldo and Ambrožič, 2018). The test statistic (2) is distributed according to the Fisher distribution Ffu,f : – T1 2 ≤ Ffu,f : the null hypothesis H0 cannot be rejected and with significance level α we cannot claim that deformations have occurred in the network. In other words, the deformations are not statisti- cally significant. – T1 2 > Ffu,f : we reject the null hypothesis H0 and claim with a significance level smaller than α that deformation has occurred in the network. In other words, the deformations are statistically significant. If compliance testing of the geodetic network confirms the presence of deformations in the network, we proceed to a more detailed consideration and localization of the displacements of points in the geodetic network. 2.3 Testing of affinity of the geodetic network The classical approach to testing of affinity of the geodetic network with the X-method is based on the for- mation of selected triangles and the calculation of the kinematic parameters (vector p) in these triangles. As a rule, such triangles are not overlapping. The choice of the vertices of the triangles has a decisive influence on the calculation of the kinematic parameters. If the points in the selected triangle move in any direction and the movements are large enough, we calculate large values of the kinematic parameters. However, if the points in the selected triangle all move in the same direction and their displacements are similar in magnitude, we usually calculate small values of the kinematic parameters and the change in the shape and size of the triangle is smaller. In the classical approach, we usually choose a limited number of triangles. From the adjusted coordinates of the 1st and 2nd sets of measurement epoch t1 and t2, we calculate the change in the measurement value which can also be written as a function of point movements (Welsch and Zhang, 1983; Soldo and Ambrožič, 2018): d l = l2 − l1 = L ⋅ u, (7) with the corresponding cofactor matrix of the change in measured value Qd l = LQu L T. (8) It is important to emphasize that the matrix of partial derivates of measurements by coordinate unknowns l L x̂ ∂ =  ∂  depends on the nature of the measurements, i.e., whether we consider only lengths, only angles, or both. In our modified method we create all possible triangles in the network to obtain more information about changes in the network. We may miss important changes due to the inappropriate selection of triangles. It is natural to calculate lengths and angles in a triangle, therefore in our proposal we first test the change of lengths, then the change of angles and only finally the change of kinematic parameters in the triangle. Since calculating all possible combinations is not a computational problem anymore, it makes sense to gather all information about changes in the network. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 136 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N 2.3.1 Testing lengths in the geodetic network The test is performed by testing changes in the datum independent lengths connecting a single point to other points. We write down the null and alternative hypothesis (Welsch, 1982): H0: E(dldDij) = 0… the length difference between the points Pi and Pj has not changed between two epochs; Ha: E(dldDij) ≠ 0… the length difference between the points Pi and Pj has changed between two epochs. Let us calculate the test statistic for the length difference between points Pi and Pj (Welsch, 1982): 1 2 2 2 , ij ij ij ij dD dD dD D D dl Q dl T n s − = ⋅ (9) where nD = 1 is the number of degrees of freedom (Welsch, 1982; Soldo and Ambrožič, 2018). The length difference is calculated with the following equation dldDij = Dij2 − Dij1, (10) where the element dldDij refers to the length 1 1 1 1 1 2 2ˆ ˆ ˆ ˆ( ) ( )ij j i j iD y y x x= − + − and 2 2 2 2 2 2 2ˆ ˆ ˆ ˆ( ) ( )ij j i j iD y y x x= − + − , calculated from the adjusted coordinates between the points Pi and Pj of the 1st and 2nd sets of measurements. The element of the cofactor matrix of the change in the value of length in equation (9) is calculated by equation (8) Qd Dij = Ld DijQud Dij L T d Dij , where the vector of partial derivatives of the measurements LdDij is written as follows (Welsch and Zhang, 1983; Soldo and Ambrožič, 2018): sin cos sin cos , ˆ ij ij dD dD ij ij ij ijν ν ν ν ∂   = = − −    ∂   l L x (11) where νij = (νij1 + νij2) ⁄ 2 is the average of the bearings νij1 and νij2, calculated from the adjusted coordin- ates between points Pi and Pj , 1 1 2 2 1 1 1 2 2 2 ˆ ˆ ˆ ˆ arctan and arctan ˆ ˆ ˆ ˆ j i j i ij ij j i j i y y y y x x x x ν ν − − = = − − in the 1st and 2nd sets of measurements. In the submatrix QudDij are the corresponding elements of the matrix Qu, which refer to points Pi and Pj. The test statistic (9) is distributed according to the Fisher distribution F1,f : – T 2 2Dij ≤ F1,f : the null hypothesis H0 cannot be rejected and with significance level α we cannot claim that the length between considered points has changed, – T 2 2Dij > F1,f : we reject the null hypothesis H0 and claim with a significance level smaller than α that the length between the considered points has changed. 2.3.2 Testing angles in the geodetic network The test is performed by testing changes in datum independent angles connecting a single point to other points. We write down the null and the alternative hypothesis: Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 137 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N H0: E(dldαijk) = 0... the angle difference between the points Pi, Pj and Pk has not changed between two epochs; Ha: E(dldαijk) ≠ 0... the angle difference between the points Pi, Pj and Pk has changed between two epochs. We create the test statistic for the angle difference between the points Pi, Pj and Pk: 1 2 2 2 , ijk ijk ijk ijk d d ddl Q dl T n s α α α α α − = ⋅ (12) where nα = 1 is the number of degrees of freedom. The angle difference is calculated with following expression dldαijk = αijk2 − αijk1, (13) where the element dldαijk refers to the angles αijk1 = νik1 − νij1 and αijk2 = νik2 − νij2 with the vertex at the point Pi against the points Pj and Pk in the 1 st and 2nd sets of measurements, where νij1 and νik1, νij2 and νik2 are the bearings from the offset coordinates between points Pi and Pj, Pi and Pk in the 1 st and 2nd sets of measurement. The element of the cofactor matrix of the change in the value of angular in equation (12) is calculated by equation (8) Qd αijk = Ld αijkQud αijk L T d αijk , where the vector of the partial derivatives of the measurements Ldαijk is written as cos sin cos sincos sin cos sin , ˆ ijk ijk d ij ij ij ijik ik ik ik d ik ij ik ij ij ij ik ikD D D D D D D D α α ν ν ν νν ν ν ν ∂          = = − + + − −             ∂            l L x (14) where νij = (νij1 + νij2) ⁄ 2 and νik = (νik1 + νik2) ⁄ 2 are the averages of bearings νij1 and νik1, νij2 and νik2. The lengths Dij = (Dij1 + Dij2) ⁄ 2 and Dik = (Dik1 + Dik2) ⁄ 2 represent the average values of the lengths obtained from the adjusted coordinates between points Pi and Pj, Pi and Pk in the 1 st and 2nd sets of measurements. In the submatrix Qudαijk are the corresponding elements of the matrix Qu referring to the points Pi, Pj and Pk. The test statistic (12) is distributed according to the Fisher distribution F1,f : – T 2 2αijk ≤ F1,f : the null hypothesis H0 cannot be rejected and with significance level α we cannot claim that the angle between considered points has changed, – T 2 2αijk > F1,f : we reject the null hypothesis H0 and claim with a significance level smaller than α that the angle between the considered points has changed. 2.3.3 Testing triangles in a geodetic network We divide the network in question into triangles and determine whether a single triangle has changed shape between the two sets of measurements and in this way explain the coordinate differences in the network. We note the null and alternative hypotheses (Welsch, 1983): x x u 00 1 2ˆ ˆ: ( ) ( ) or ( )ijkH E E E ∆= = ... the coordinates of the points of the triangle have not changed between two epochs; x x u 01 2ˆ ˆ: ( ) ( ) or ( )ijkaH E E E ∆≠ ≠ ... the coordinates of at least one point in the triangle have changed between two epochs. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 138 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Let us calculate a test statistic for the vertices Pi, Pj and Pk of the triangle: 1 2 2 2 , ijk ijk ijk T n s ∆ ∆ − ∆ ∆ = ⋅ T uu Q u (15) where n∆ = 3 is the number of degrees of freedom (Welsch, 1983; Welsch and Zhang, 1983; Soldo and Ambrožič, 2018). We calculate the vector of displacements of the vertices in the triangle ijk∆ u by the equation (3), where x1ˆ and x 2ˆ are the vectors of the adjusted coordinates of vertices Pi, Pj and Pk in the triangle after the 1 st and 2nd sets of measurements have been adjusted. The cofactor matrix of the coordinate differences Qu is calculated by equation (4), where x xQ 1 1ˆ ˆ and x xQ 2 2ˆ ˆ are the cofactor matrices of the unknown coordinates of the vertices Pi, Pj and Pk in the triangle after the adjustment of the geodetic network. The test statistic (15) is distributed according to the Fisher distribution F1,f : – T 2 2∆ijk ≤ F3,f : the null hypothesis H0 cannot be rejected and with the significance level α we cannot claim that the coordinates in the triangle with vertices Pi, Pj and Pk have changed, – T 2 2∆ijk > F3,f : we reject the null hypothesis H0 and claim with a significance level smaller than α that the coordinates in the triangle with vertices Pi, Pj and Pk have changed. In the following, we can calculate the kinematic parameters in each triangle: p = Hu -1 . u, (16) where Hu is the deformation model matrix that relates the kinematic parameters to the displacements of the points in the triangle uH ˆ ˆ ˆ0 1 0 ˆ ˆ ˆ0 0 1 ˆ ˆ ˆ0 1 0 ˆ ˆ ˆ0 0 1 ˆ ˆ ˆ0 1 0 ˆ ˆ ˆ0 0 1 i i i i i i j j j j j j k k k k k k x y y x y x x y y x y x x y y x y x −         − =       −      , (17) PT xx xy yy x ye e e t tω =   (18) and p is a vector of kinematic parameters, where exx and eyy are the normal deformations in the direction of the coordinate axes x and y, exy = eyx is the shear deformation (only exx, eyy and exy are the deformation parameters), ω represents rotation, tx and ty are the translations in the direction of the coordinate axes x and y. Based on the basic parameters, we calculate other kinematic parameters (Welsch, 1983; Mihailović and Aleksić, 1994; Ašanin, 1986; Acar, 2010; Labant et al., 2014; Sušić et al., 2015b; Boresi and Sidebottom, 1985): Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 139 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N ∆ = exx + eyy represents the change of area, e1 = ½ (exx + eyy + ee) is the principal (the largest) normal strain, e2 = ½ (exx + eyy − ee) is the principal (the smallest) normal strain, where ee2 = (exx − eyy) 2 + 4e2xy, 1 2 I 2 e e e − = is the principal shear strain and γ = 2exy is the engineering shear strain and represents the change of the right angle between x and y directions. The bearings of the principal normal strains are calculated with the expression 2 tan 2 xy xx yy e e e ϑ = − and the bearing of the principal shear strain is calculated with the expression Ψ = ϑ + 45°. 3 CALCULATION EXAMPLE For a simple and direct comparison of the discussed modified method of deformation analysis according to the Munich approach with other methods of deformation analysis, we choose the example of a simulated geodetic network, calculated with other methods of deformation analysis (Ambrožič, 2001; Ambrožič, 2004; Marjetič et al., 2012; Vrečko and Ambrožič, 2013; Soldo and Ambrožič, 2018; Ambrožič et al., 2019; Hamza et al., 2020; Batilović et al., 2022). In the cited literature, we can see the sketch of the network and obtain all the necessary data for adjustment. Briefly summarizing the results of free network adjustment by reducing the orientation unknowns of the 1st set of measurements, we obtain s1  =  0.9699, f1 = 30 and the 2 nd set of measurements s2 = 1.1562 and f2 = 30. Since we confirm the identity of the geodetic datum and the homogeneity of the accuracy in the considered term measurements, we calculate the total reference variance a posteriori s2 = 1.1387 and the total number of degrees of freedom is f = 60 (equation 1). In addition, a significance level α = 5 % is chosen for all. The test of congruence of the geodetic network by the classical approach and by the proposed modified method of deformation analysis is exactly the same, so we write only that T1 2 = 141.29 (equation 2), which exceeds the critical value at the chosen test characteristic F11,60 = 1.95. Therefore, we reject the null hypothesis and claim with a significance level smaller than α = 5 %, that deformations occurred in the network, and continue the deformation analysis by testing of affinity of the geodetic network. The deformation analysis according to the proposed modified method is performed by testing of affin- ity of the geodesic network by first considering all the lengths in the geodetic network, of which there are 21. For each calculated length we create a test statistic T2 2 Dij (equation 9) and compare it with the critical value at the chosen level of test characteristics F1,60 = 4.00, the results are given in Table 1. We calculate the actual risk αT for rejecting the null hypothesis, which we compare with the significance level α = 5  %. In the cases where the actual risk is lower than the chosen one, we must reject the null hypothesis and claim that the deformations are statistically significant. However, in the opposite cases, we can conclude that the lengths between stationary points have not changed statistically significantly, since the actual risk is greater than 5 % in all cases. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 140 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Table 1: Analysis of testing all lengths between the points. Length Between the points T 22Dij H0 αT [%] Length Between the points T 22Dij H0 αT [%] 1 1-2 19.89 reject 0.00* 12 3-4 124.81 reject 0.00* 2 1-3 49.38 reject 0.00* 13 3-5 62.37 reject 0.00* 3 1-4 87.04 reject 0.00* 14 3-6 10.41 reject 0.20 4 1-5 64.84 reject 0.00* 15 3-7 9.20 reject 0.36 5 1-6 10.05 reject 0.24 16 4-5 0.08 not rej. 77.85 6 1-7 689.26 reject 0.00* 17 4-6 0.01 not rej. 91.54 7 2-3 109.75 reject 0.00* 18 4-7 186.39 reject 0.00* 8 2-4 84.02 reject 0.00* 19 5-6 0.63 not rej. 42.87 9 2-5 163.39 reject 0.00* 20 5-7 83.12 reject 0.00* 10 2-6 113.96 reject 0.00* 21 6-7 77.68 reject 0.00* 11 2-7 113.61 reject 0.00* * actual risk is smaller than 0.005 % Figure 1: Analysis of the lengths between the points: the lengths where the null hypothesis cannot be rejected are marked in red and the lengths where the null hypothesis is rejected are marked in black. Table 1 and Figure 1 show that only the lengths between points 4-5, 4-6 and 5-6 did not change sta- tistically significantly. Then we continue the deformation analysis using the proposed modified method by testing of affinity of the geodetic network considering all the angles in the geodetic network, of which there are 105. For each calculated angle, we create the test statistic T2 2 αijk (equation 12) and compare it with the critical value at the chosen level of test characteristics F1,60 = 4.00, the results are shown in Table 2. We calculate the actual risk αT for rejecting the null hypothesis, which we compare with the significance level α = 5  %. When testing angles, it is not possible to unambiguously determine which angle does not change its value, since we also test angles between two fixed and one moving points, or even between one fixed and two moving points. From Table 2, we can conclude that the actual risk is greater than 5 % for all combinations of angles where all three stationary points occur, such as angles 4-5-6 (αT = 33 %), 5-4-6 (αT = 44 %), and 6-4-5 (αT = 74 %). Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 141 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Table 2: Analysis of testing all angles between the points. Angle Between the points T 22αijk H0 αT [%] Angle Between the points T 22αijk H0 αT [%] 1 1-2-3 765.25 reject 0.00* 36 3-2-4 172.51 reject 0.00* 2 1-2-4 382.51 reject 0.00* 37 3-2-5 256.93 reject 0.00* 3 1-2-5 377.49 reject 0.00* 38 3-2-6 322.97 reject 0.00* 4 1-2-6 409.63 reject 0.00* 39 3-2-7 15.28 reject 0.02 5 1-2-7 304.97 reject 0.00* 40 3-4-5 2.50 not rej. 11.88 6 1-3-4 55.84 reject 0.00* 41 3-4-6 4.70 reject 3.42 7 1-3-5 2.08 not rej. 15.41 42 3-4-7 183.15 reject 0.00* 8 1-3-6 38.50 reject 0.00* 43 3-5-6 2.24 not rej. 14.01 9 1-3-7 32.56 reject 0.00* 44 3-5-7 368.79 reject 0.00* 10 1-4-5 24.42 reject 0.00* 45 3-6-7 342.61 reject 0.00* 11 1-4-6 104.38 reject 0.00* 46 4-1-2 153.77 reject 0.00* 12 1-4-7 0.33 not rej. 56.77 47 4-1-3 7.99 reject 0.64 13 1-5-6 78.35 reject 0.00* 48 4-1-5 0.05 not rej. 83.07 14 1-5-7 20.96 reject 0.00* 49 4-1-6 3.88 not rej. 5.36 15 1-6-7 101.10 reject 0.00* 50 4-1-7 33.76 reject 0.00* 16 2-1-3 844.94 reject 0.00* 51 4-2-3 17.86 reject 0.01 17 2-1-4 474.80 reject 0.00* 52 4-2-5 42.39 reject 0.00* 18 2-1-5 448.55 reject 0.00* 53 4-2-6 69.33 reject 0.00* 19 2-1-6 406.03 reject 0.00* 54 4-2-7 22.08 reject 0.00* 20 2-1-7 916.07 reject 0.00* 55 4-3-5 4.18 reject 4.52 21 2-3-4 361.72 reject 0.00* 56 4-3-6 3.20 not rej. 7.89 22 2-3-5 426.83 reject 0.00* 57 4-3-7 0.28 not rej. 59.82 23 2-3-6 456.29 reject 0.00* 58 4-5-6 0.98 not rej. 32.59 24 2-3-7 76.78 reject 0.00* 59 4-5-7 14.57 reject 0.03 25 2-4-5 69.08 reject 0.00* 60 4-6-7 15.39 reject 0.02 26 2-4-6 132.49 reject 0.00* 61 5-1-2 1.17 not rej. 28.29 27 2-4-7 37.28 reject 0.00* 62 5-1-3 4.23 reject 4.40 28 2-5-6 68.82 reject 0.00* 63 5-1-4 5.51 reject 2.23 29 2-5-7 230.59 reject 0.00* 64 5-1-6 5.41 reject 2.34 30 2-6-7 365.41 reject 0.00* 65 5-1-7 230.42 reject 0.00* 31 3-1-2 561.33 reject 0.00* 66 5-2-3 15.02 reject 0.03 32 3-1-4 0.39 not rej. 53.44 67 5-2-4 4.47 reject 3.86 33 3-1-5 1.06 not rej. 30.63 68 5-2-6 1.15 not rej. 28.79 34 3-1-6 11.15 reject 0.14 69 5-2-7 232.58 reject 0.00* 35 3-1-7 468.25 reject 0.00* 70 5-3-4 38.75 reject 0.00* Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 142 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Angle Between the points T 22αijk H0 αT [%] Angle Between the points T 22αijk H0 αT [%] 71 5-3-6 9.61 reject 0.30 89 6-4-7 127.84 reject 0.00* 72 5-3-7 240.57 reject 0.00* 90 6-5-7 90.46 reject 0.00* 73 5-4-6 0.60 not rej. 44.04 91 7-1-2 132.04 reject 0.00* 74 5-4-7 73.79 reject 0.00* 92 7-1-3 199.15 reject 0.00* 75 5-6-7 106.42 reject 0.00* 93 7-1-4 9.90 reject 0.26 76 6-1-2 296.34 reject 0.00* 94 7-1-5 41.02 reject 0.00* 77 6-1-3 35.57 reject 0.00* 95 7-1-6 33.70 reject 0.00* 78 6-1-4 98.75 reject 0.00* 96 7-2-3 27.83 reject 0.00* 79 6-1-5 69.53 reject 0.00* 97 7-2-4 44.72 reject 0.00* 80 6-1-7 314.80 reject 0.00* 98 7-2-5 233.68 reject 0.00* 81 6-2-3 258.54 reject 0.00* 99 7-2-6 207.06 reject 0.00* 82 6-2-4 38.20 reject 0.00* 100 7-3-4 215.43 reject 0.00* 83 6-2-5 22.19 reject 0.00* 101 7-3-5 432.37 reject 0.00* 84 6-2-7 20.63 reject 0.00* 102 7-3-6 332.44 reject 0.00* 85 6-3-4 97.78 reject 0.00* 103 7-4-5 129.96 reject 0.00* 86 6-3-5 31.15 reject 0.00* 104 7-4-6 57.94 reject 0.00* 87 6-3-7 321.56 reject 0.00* 105 7-5-6 5.96 reject 1.76 88 6-4-5 0.11 not rej. 74.42 * actual risk is smaller than 0.005 % Figure 2: Analysis of angles between the points: angles are marked in red where the null hypothesis cannot be rejected and the angles are marked in black where the null hypothesis is rejected. Table 2 and Figure 2 show that some angles between points did not change statistically significantly. The deformation analysis is continued according to the proposed modified method by testing of affinity of the geodesic network by considering the consistency of all triangles in the geodetic network, of which there are 35. For each calculated triangle, we create a test statistic T2 2 ∆ijk (equation 15) and compare it with the critical value at the chosen level characteristics of the test F3,60 = 2.76, the results are given in Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 143 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Table 3. We calculate the actual risk αT for rejecting the null hypothesis, which we compare with the significance level α = 5 %. We can conclude that the triangle between stationary points 4-5-6 did not change statistically significantly, since the actual risk is much higher than 5 % (αT = 77 %). Table 3: Analysis of the triangle congruence test between the points. Triangle Between the points T 22∆ijk H0 αT [%] Triangle Between the points T 22∆ijk H0 αT [%] 1 1-2-3 286.17 reject 0.00* 19 2-3-7 62.03 reject 0.00* 2 1-2-4 159.09 reject 0.00* 20 2-4-5 60.86 reject 0.00* 3 1-2-5 157.59 reject 0.00* 21 2-4-6 68.60 reject 0.00* 4 1-2-6 166.60 reject 0.00* 22 2-4-7 110.69 reject 0.00* 5 1-2-7 336.48 reject 0.00* 23 2-5-6 67.44 reject 0.00* 6 1-3-4 62.88 reject 0.00* 24 2-5-7 147.62 reject 0.00* 7 1-3-5 42.33 reject 0.00* 25 2-6-7 133.31 reject 0.00* 8 1-3-6 24.16 reject 0.00* 26 3-4-5 42.87 reject 0.00* 9 1-3-7 278.24 reject 0.00* 27 3-4-6 41.92 reject 0.00* 10 1-4-5 35.00 reject 0.00* 28 3-4-7 116.79 reject 0.00* 11 1-4-6 46.79 reject 0.00* 29 3-5-6 20.86 reject 0.00* 12 1-4-7 271.23 reject 0.00* 30 3-5-7 162.34 reject 0.00* 13 1-5-6 33.96 reject 0.00* 31 3-6-7 140.22 reject 0.00* 14 1-5-7 249.07 reject 0.00* 32 4-5-6 0.37 not rej. 77.30 15 1-6-7 229.98 reject 0.00* 33 4-5-7 95.96 reject 0.00* 16 2-3-4 136.82 reject 0.00* 34 4-6-7 98.21 reject 0.00* 17 2-3-5 158.79 reject 0.00* 35 5-6-7 56.68 reject 0.00* 18 2-3-6 163.70 reject 0.00* * actual risk is smaller than 0.005 % We do not show the image of the triangles between the points because the figure becomes unlegible due to a large number of triangles. Due to testing a large number of all lengths between points in the network, an even larger number of all angles between points in the network, and also a large number of testing of congruence of all possible triangles in the network, which is the essence of the proposed modified method of deformation analysis, the analysis is difficult to represent. We can simplify the analysis and make it more legible by performing all three testing of quantities per unit, i.e. per individual triangle. Table 4 shows the results of testing all three quantities in a single triangle. Table 4: Analysis of testing all three quantities in triangles. Triangle Between the points i-j-k H0-D i-j j-k i-k H0-α i-j-k j-i-k k-i-j H0-∆ i-j-k 1 1-2-3 reject reject reject reject reject reject reject 2 1-2-4 reject reject reject reject reject reject reject 3 1-2-5 reject reject reject reject reject not reject reject Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 144 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Triangle Between the points i-j-k H0-D i-j j-k i-k H0-α i-j-k j-i-k k-i-j H0-∆ i-j-k 4 1-2-6 reject reject reject reject reject reject reject 5 1-2-7 reject reject reject reject reject reject reject 6 1-3-4 reject reject reject reject not reject reject reject 7 1-3-5 reject reject reject not rej. not rej. reject reject 8 1-3-6 reject reject reject reject reject reject reject 9 1-3-7 reject reject reject reject reject reject reject 10 1-4-5 reject not reject reject reject not reject reject reject 11 1-4-6 reject not reject reject reject not reject reject reject 12 1-4-7 reject reject reject not reject reject reject reject 13 1-5-6 reject not reject reject reject reject reject reject 14 1-5-7 reject reject reject reject reject reject reject 15 1-6-7 reject reject reject reject reject reject reject 16 2-3-4 reject reject reject reject reject reject reject 17 2-3-5 reject reject reject reject reject reject reject 18 2-3-6 reject reject reject reject reject reject reject 19 2-3-7 reject reject reject reject reject reject reject 20 2-4-5 reject not reject reject reject reject reject reject 21 2-4-6 reject not reject reject reject reject reject reject 22 2-4-7 reject reject reject reject reject reject reject 23 2-5-6 reject not reject reject reject not reject reject reject 24 2-5-7 reject reject reject reject reject reject reject 25 2-6-7 reject reject reject reject reject reject reject 26 3-4-5 reject not reject reject not reject reject reject reject 27 3-4-6 reject not reject reject reject not reject reject reject 28 3-4-7 reject reject reject reject not reject reject reject 29 3-5-6 reject not reject reject not reject reject reject reject 30 3-5-7 reject reject reject reject reject reject reject 31 3-6-7 reject reject reject reject reject reject reject 32 4-5-6 not rej. not rej. not rej. not rej. not rej. not rej. not rej. 33 4-5-7 not reject reject reject reject reject reject reject 34 4-6-7 not reject reject reject reject reject reject reject 35 5-6-7 not reject reject reject reject reject reject reject Figure 3 does not show the kinematic parameters for each triangle, but the differences of the adjusted coordinates between two epochs, which is one of the results of the deformation analysis according to the Munich method (Welsch and Zhang, 1983; Welsch, 1983; Soldo and Ambrožič, 2018; Sušić et al., 2015b; Sušić et al., 2017; Mihailović and Aleksić, 1994). There are too many triangles, and some of them are very elongated, so the image would be very illegible. Based on the test results presented in Table 4, it can be quickly observed that all lengths, all angles and congruence in triangle 32 did not change statistically significantly, which is shown in Figure 3 by plotting Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 145 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N the displacements. With standard significance level α = 5 %, we claim that the points of this triangle have not shifted significantly, since we tested the triangle both in terms of changes in the lengths and angles in the triangle, and in the coherence of the entire triangle. Figure 3: Illustration of point displacements: the movements at points 4, 5 and 6 are marked in red if the null hypothesis cannot be rejected and if the null hypothesis is rejected, they are marked in black. According to equation (18) we calculate the basic kinematic parameters and show them in Table 5. Other deformation parameters are not calculated and displayed, as they can be easily calculated with the above equations. Table 5: Calculated kinematic parameters in all triangles. Triangle Between the points i-j-k exx [μs] exy [μs] eyy [μs] ω [''] tx [mm] ty [mm] 1 1-2-3 -161.27 82.79 -18.70 -2.7 0.021 -0.062 2 1-2-4 -43.92 57.70 -18.70 -7.9 -0.096 -0.012 3 1-2-5 11.14 50.49 -18.70 -9.4 -0.151 0.003 4 1-2-6 151.73 50.35 -18.70 -9.4 -0.292 0.003 5 1-2-7 46.19 76.48 -18.70 -4.0 -0.186 -0.049 6 1-3-4 52.09 -22.84 32.64 0.3 -0.072 -0.022 7 1-3-5 24.18 -4.11 20.39 0.9 -0.060 -0.031 8 1-3-6 39.05 5.69 4.66 4.6 -0.066 -0.044 9 1-3-7 158.66 -16.92 -7.75 13.8 -0.119 -0.053 10 1-4-5 21.33 8.38 2.64 -1.2 -0.080 -0.016 11 1-4-6 43.04 0.08 -10.54 2.7 -0.074 -0.013 12 1-4-7 368.00 -113.88 -233.38 102.9 0.007 0.031 13 1-5-6 26.76 -12.01 -18.44 3.5 -0.042 0.003 14 1-5-7 2.38 79.05 85.25 17.9 -0.212 -0.088 15 1-6-7 86.78 33.95 13.46 0.6 -0.162 -0.061 16 2-3-4 -14.58 -58.59 75.42 13.5 0.315 -0.187 17 2-3-5 -62.75 -9.55 36.67 8.7 0.218 -0.136 Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 146 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Triangle Between the points i-j-k exx [μs] exy [μs] eyy [μs] ω [''] tx [mm] ty [mm] 18 2-3-6 -98.67 29.36 0.77 5.6 0.146 -0.088 19 2-3-7 -54.19 -0.77 -8.92 13.2 0.235 -0.075 20 2-4-5 -32.33 12.71 4.05 0.8 0.066 -0.054 21 2-4-6 -34.60 22.41 -13.45 -0.7 0.034 -0.021 22 2-4-7 -28.08 1.59 -68.24 5.1 0.126 0.081 23 2-5-6 -21.31 18.08 -18.57 -2.7 0.011 0.002 24 2-5-7 -129.06 -193.66 -434.50 -1.9 0.550 1.042 25 2-6-7 13.88 58.63 6.90 3.0 -0.050 -0.116 26 3-4-5 81.75 -12.17 -2.07 -6.7 -0.244 0.112 27 3-4-6 55.49 -26.77 13.21 -1.6 -0.092 0.053 28 3-4-7 20.50 -57.18 0.71 3.0 0.111 0.102 29 3-5-6 14.21 -15.96 11.03 -0.5 -0.028 0.029 30 3-5-7 -83.15 -50.90 0.43 2.3 0.284 0.096 31 3-6-7 1279.93 35.50 -26.54 45.7 -1.984 -0.397 32 4-5-6 -5.80 0.43 1.32 -2.3 -0.006 0.020 33 4-5-7 -57.45 -17.02 -1.65 -4.8 0.135 0.101 34 4-6-7 -119.44 -3.87 34.03 14.9 0.153 0.112 35 5-6-7 -39.65 11.85 25.68 -8.7 0.031 0.041 4 CONCLUSIONS In previous research, we have demonstrated that the results are very similar regardless of which method of deformation analysis is used for the test case (Ambrožič, 2001; Ambrožič, 2004; Marjetič et al., 2012; Vrečko and Ambrožič, 2013; Soldo and Ambrožič, 2018; Ambrožič et al., 2019; Hamza et al., 2020; Batilović et al., 2022). In our research, we focused on important improvements that would allow a more reliable determination of the deformations of the object under consideration or the measured surface. In the proposed modified method of deformation analysis according to the Munich approach, we propose that before the phase of testing of affinity of the geodetic network, therefore before the formation of triangles in the network, in addition to testing changes in independent lengths, we also perform testing of independent angles between all points in the network. In the next phase, we form all possible triangles between the points of the network and not only the selected ones, because in this way we obtain more reliable information about possible changes in the shape of the triangle and, consequently, about the points that have shifted in a statistically significant way. When analyzing the test of all lengths in the network, with the proposed method, we managed to find the lengths between points 4-5, 4-6 and 5-6, that did not change statistically significantly, so the null hypoth- esis cannot be rejected (Table 1 and Figure 1). This is consistent with the simulated network movements when points 4, 5 and 6 are presumed stable. When analyzing the test of all angles in the network using the proposed method, we managed to find all combinations of angles between points 4-5-6, 5-4-6 and 6-4-5 that did not change statistically significantly, so the null hypothesis cannot be rejected (Table 2 and Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 147 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Figure 2). Analyzing the consistency test of all triangles in the network, we managed to find one triangle with vertices in points 4, 5 and 6 that did not change statistically significantly, so the null hypothesis cannot be rejected (Table 3). If we write down the results of the tests more transparently, we can find that all lengths, all angles and the congruence in triangle 32, i.e. between points 4-5-6, did not change statistically significantly (Table 4). We conclude that with the significance level α = 5 % we cannot claim that the points of this triangle moved, which is consistent with the simulated movements in the network. The advantages of the proposed modified method are that the user of the deformation analysis usually forms: – triangles between adjacent points, but we propose between all points and thus obtain information abo- ut the deformations between all points, which can be overlooked if we form the triangles themselves. – "nice" shapes of triangles, which in turn causes us to lose information about the deformations of such triangle in the case of a very elongated triangle or a triangle where one side is much shorter than the other two. The disadvantage of the proposed modified method is in the case of a geodetic network when only two points are stationary. In this case, we cannot form a triangle (with three points) if all three tested quantities (lengths, angles and congruence of the triangle) would not change statistically significantly. According to the proposed method, we could conclude that there are too few stable points. We estimate that the proposed modified method contains important improvements that help us to find more reliably stable points or statistically characteristic deformations of the considered object, which is the essence of any deformation analysis. Considering the advanced capabilities of computational oper- ations and storage of computational results, we conclude that the spatial and temporal complexity of the proposed algorithm is not problematic. Acknowledgments The paper was prepared within the framework of the research program P2-0227: Geoinformation Infrastructure and Sustainable Spatial Development of Slovenia and basic research project J2-2489: SLOKIN – Geokinematic Model of Slovenian Territory which are funded by the Public Agency for Research Activities of the Republic of Slovenia. Literature and references Acar, M. (2010). Determination of strain accumulation in landslide areas with GPS measurements. Scientific Research and Essays, 5 (8), 763-768. 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(ur.), Deformationsanalysen ’83, Geometrische Analyse und Interpretation von Deformationen Geodätischer Netze. Beiträge zum Geodätischen Seminar. Schriftenreihe des Wissenschaftlichen Studiengangs Vermessungswesen der Hochschule der Bundeswehr, Heft 9, München, pp. 299-328. Welsch, W.M. (1986). Some aspects of the analysis of geodetic strain observations in kinematic models. Tectonophysics, 39 (1-4), 437-458. DOI: https://doi. org/10.1016/0040-1951(86)90132-0. Savšek S., Ambrožič T. (2023). Modified method of deformation analysis according to the Munich approach. Geodetski vestnik, 67 (2), 131-164. DOI: https://doi.org/10.15292/geodetski-vestnik.2023.02.131-164 Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 149 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N 1 UVOD Spremljanje stabilnosti gradbeno-inženirskih ali naravnih objektov, tako imenovani monitoring, je sistematično merjenje in sledenje spremembam oblike in/ali dimenzije obravnavanega objekta kot posledica napetosti, ki jih povzročajo obremenitve. Kontinuirano spremljanje deformacij z beleženjem izmerjenih vrednosti je osnovni pogoj za analizo deformacij, na podlagi katere pravočasno predlagamo vzdrževanje ali sanacijo obravnavanega objekta. Spremljanje premikanja je primarno povezano s področjem inženirske geodezije, kjer lahko uporabimo različne merilne naprave ali senzorje. Za ustrezno analizo deformacij so bili razviti različni postopki deformacijske analize (Chrzanowski et al., 1986; Ašanin, 1986; Mihailović in Aleksić, 1994). V prispevku obravnavamo deformacijsko analizo po postopku München. Razvil jo je W. M. Welsch, ki je deloval na Inštitutu za geodezijo v okviru Visoke vojaške šole v Münchnu. Metoda temelji na analizi deformacij trikotnikov, katerih oglišča tvorijo geodetske točke v mreži in katerih spremembe koordinat izračunamo na osnovi spremembe kotov in dolžin v mreži. Deformacijsko analizo po postopku München lahko izvedemo na dva načina (Welsch, 1983; Welsch in Zhang, 1983; Soldo in Ambrožič, 2018): – z metodo X, ki temelji na primerjavi koordinat identičnih točk geodetske mreže dveh terminskih izmer. Ker so koordinate točk odvisne od geodetskega datuma, je pomembno, da se rezultati izrav- nave proste mreže nanašajo na isti geodetski datum. Če se število točk v mreži v terminskih izmerah razlikuje, izločimo koordinatne neznanke neidentičnih točk s transformacijo S (van Mierlo, 1987); – z metodo L, ki temelji na primerjavi količin, neodvisnih od geodetskega datuma, to so koti in dolžine. Klasični pristop deformacijske analize po postopku München zahteva izravnavo geodetske mreže kot proste mreže za vsako terminsko izmero z zagotovitvijo identičnega geodetskega datuma ter testiranje homogenosti natančnosti meritev obravnavanih izmer. V nadaljevanju se izvede testiranje skladnosti geodetske mreže, s katerim skušamo ugotoviti, ali je prišlo do premikov in deformacij celotnega objekta. Če ugotovimo statistično značilne deformacije objekta, nadaljujemo s testiranjem preoblikovanja geodetske mreže, pri čemer mrežo razdelimo na izbrane trikotnike in ugotavljamo spremembo oblike posameznega trikotnika. Izračunamo osnovne kinematične parametre v posameznem trikotniku in nato še druge parametre, kot so normalne in strižne deformacije. Nazad- nje izvedemo še testiranje sprememb datumsko neodvisnih dolžin v mreži kot predlaga avtor postopka W. M. Welsch, in tako odkrijemo točke, ki so se statistično značilno premaknile (Welsch in Zhang, 1983; Ašanin, 1986). Modificiran pristop, ki ga predlagamo v prispevku, temelji na tvorjenju vseh možnih trikotnikov v mreži, ne le izbranih, ter poleg testiranja sprememb datumsko neodvisnih dolžin vključuje tudi testiranje neod- visnih kotov v vseh trikotnikih, ki jih tvorimo. Tako zajamemo bistveno več informacij o spremembah v mreži in z večjo gotovostjo najdemo točke, ki so se značilno premaknile, kar je pomembna izboljšava klasičnega pristopa deformacijske analize po postopku München. MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN OSNOVNE INFORMACIJE O ČLANKU: GLEJ STRAN 131 Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 150 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI| EN Z obema načinoma deformacijske analize po postopku München, torej po metodi X in po metodi L, dobimo v fazi testiranja skladnosti geodetske mreže in v fazi testiranja preoblikovanja geodetske mreže popolnoma enake rezultate (Welsch, 1983; Welsch in Zhang, 1983; Soldo in Ambrožič, 2018), zato bomo za testiranje skladnosti in testiranje preoblikovanja geodetske mreže v predlaganem modificiranem pristopu uporabili le metodo X. 2 DEFORMACIJSKA ANALIZA PO POSTOPKU MÜNCHEN 2.1 Osnovni pogoji za izvedbo deformacijske analize Za izvedbo deformacijske analize je pomembno, da zagotovimo kakovost in skladnost natančnosti meritev v terminskih izmerah. To izvedemo tako, da preverimo homogenost natančnosti meritev v terminskih izmerah z ničelno domnevo H0: E(s1 2) = E(s2 2), ki pravi, da sta natančnosti meritev v terminskih izmerah enaki, kjer sta si a posteriori referenčni standardni deviaciji enote uteži za posamezno terminsko izmero. Iz meritev moramo izločiti grobo pogrešene meritve po uveljavljenih postopkih, kot so Baardova, Popeova, danska ali ustrezna druga metoda (Caspary, 1988; Grigillo in Stopar, 2003; Vrce, 2011). Metode deformacijske analize zahtevajo izravnavo meritev posamezne terminske izmere kot prosto mrežo z minimalno sledjo matrike kofaktorjev neznank (Ambrožič, 2001; Sušić et al., 2017). Pomembno je, da obravnavamo le koordinatne neznanke, zato orientacijske neznanke izločimo z redukcijo neznank v enačbah popravkov. Če obravnavamo le dolžinske meritve, moramo reducirati tudi neznanko zaradi faktorja merila v mreži (Van Mierlo, 1978). Deformacijsko analizo lahko izvedemo le na identičnih točkah v mreži, zato s transformacijo S po potrebi izločimo koordinatne neznanke neidentičnih točk v obravnavanih terminskih izmerah (Van Mierlo, 1978; Caspary 1988; Marjetič in Stopar, 2007; Sušić et al., 2015a; Marković et al., 2019). Rezultat izravnave sta ocenjena vektorja izravnanih koordinat točk ˆ ix s pripadajočima matrikama kofaktorjev koordinatnih neznank x xQ ˆ ˆi i ter referenčni varianci a posteriori enote uteži si 2 za posamezno terminsko izmero. Ko zagotovimo identični geodetski datum in preverimo homogenost natančnosti v obravnavanih ter- minskih izmerah, izračunamo referenčno varianco a posteriori z izrazom 1 2 2 2 1 1 2 22 1 1 2 2 1 2 , f s f s s f f f + + = = + T T ll llv P v v P v (1) kjer sta v1 in v2 vektorja popravkov meritev, Pll1 in Pll2 matriki uteži meritev, f1 in f2 števili nadštevilnih meritev in velja f1 + f2 = f , s1 2 in s2 2 referenčni varianci a posteriori po izravnavi predhodne in tekoče terminske izmere t1 in t2. Če v postopku testiranja skladnosti geodetske mreže ugotovimo statistično značilne deformacije, na- daljujemo s klasičnim pristopom testiranja preoblikovanja geodetske mreže, tako da mrežo delimo na trikotnike in ugotavljamo, ali je posamezen trikotnik značilno spremenil obliko med dvema termin- skima izmerama. 2.2 Testiranje skladnosti geodetske mreže Odločitev, ali so se v geodetski mreži pojavili premiki in deformacije, opravimo s testiranjem skladnosti. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 151 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Sestavimo ničelno in alternativno hipotezo (Welsch, 1983): x x u 00 1 2ˆ ˆ: ( ) ( ) oz. ( )H E E E= = … koordinate vseh točk v mreži se med dvema terminskima izmerama niso spremenile; x x u 01 2ˆ ˆ: ( ) ( ) oz. ( )aH E E E≠ ≠ … koordinate vsaj ene točke v mreži so se med dvema terminskima izmerama spremenile. Pri testiranju skladnosti z metodo X z globalnim testom stabilnosti vseh točk mreže primerjamo varianco razlik koordinat točk su 2 z oceno referenčne variance a posteriori s2. Sestavimo testno statistiko: u u u u Q u2 T 12 1 2 2 , s T s f s − = = ⋅ (2) Vektor premikov identičnih točk izračunamo z naslednjim izrazom: u x x2 1ˆ ˆ ,= − (3) kjer sta x x1 2ˆ ˆin vektorja izravnanih koordinat vseh točk po izravnavi predhodne in tekoče terminske izmere t1 in t2. Matriko kofaktorjev koordinatnih razlik zapišemo kot u x x x xQ Q Q1 1 2 2ˆ ˆ ˆ ˆ ,= + (4) kjer sta x x x xQ Q1 1 2 2ˆ ˆ ˆ ˆin matriki kofaktorjev koordinatnih neznank po izravnavi, ki ju izračunamo z izrazoma x x ll x x llQ B P B G G Q B P B G G1 1 1 2 2 2 T T 1 T T 1 ˆ ˆ ˆ ˆ1 1 2 2( ) in ( ) − −= + = + . B1 in B2 sta matriki koeficientov enačb poprav- kov, Pll1 in Pll2 pa sta matriki uteži predhodne in tekoče terminske izmere v trenutkih t1 in t2 (Kuang, 1996). Datumska matrika GT ima v 2D-geodetski triangulacijski mreži obliko (Welsch, 1986): T 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 1 i i j j m m i i j j m m x y x y x y y x y x y x ...     ...  =   − − ... −   ...    G , (5) kjer so yi 0, xi 0, yj 0, xj 0, ym 0, xm 0 približne koordinate m točk geodetske mreže. Število prostostnih stopenj je: fu = u − d. (6) Z m označimo število vseh točk v geodetski mreži. V enačbi (6) u = 2m predstavlja število koordinatnih neznank v 2D-mreži in d defekt datuma geodetske mreže (Welsch, 1983; Welsch in Zhang, 1983; Soldo in Ambrožič, 2018). Testna statistika (2) se porazdeljuje po Fisherjevi porazdelitvi Ffu,f : Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 152 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI| EN – T1 2 ≤ Ffu,f : ničelne hipoteze H0 ne moremo zavrniti in s tveganjem α ne moremo trditi, da so se v mreži pojavile deformacije. Z drugimi besedami, deformacije niso statistično značilne. – T1 2 > Ffu,f : ničelno hipotezo H0 zavrnemo in s tveganjem, manjšim od α, trdimo, da so se v mreži pojavile deformacije. Z drugimi besedami, deformacije so statistično značilne. Če s testiranjem skladnosti geodetske mreže potrdimo prisotnost deformacij v mreži, nadaljujemo s podrobnejšo obravnavo in lociranjem premikov točk v geodetski mreži. 2.3 Testiranje preoblikovanja geodetske mreže Klasični pristop testiranja preoblikovanja mreže z metodo X sloni na tvorjenju izbranih trikotnikov in računanju kinematičnih parametrov (vektor p) v teh trikotnikih. Navadno izberemo trikotnike, ki se med seboj ne prekrivajo. Izbira oglišč trikotnikov pa ključno vpliva na izračun kinematičnih parametrov. Če se točke v izbranem trikotniku premaknejo v poljubno smer in so premiki dovolj veliki, izračunamo velike vrednosti kinematičnih parametrov. Če pa se točke v izbranem trikotniku premaknejo vse v isto smer in je njihov premik podobno velik, navadno izračunamo male vrednosti kinematičnih parametrov in je sprememba oblike in velikosti trikotnika manjša. Pri klasičnem pristopu navadno izberemo omejeno število trikotnikov. Iz izravnanih koordinat točk predhodne in tekoče terminske izmere t1 in t2 izračunamo spremembo vrednosti meritev, ki jo lahko zapišemo tudi kot funkcijo premikov točk (Welsch in Zhang, 1983; Soldo in Ambrožič, 2018): d l = l2 − l1 = L ⋅ u, (7) s pripadajočo matriko kofaktorjev spremembe vrednosti meritev Qd l = LQu L T. (8) Pomembno je poudariti, da je matrika parcialnih odvodov meritev po koordinatnih neznankah l L x̂ ∂ =  ∂  odvisna od vrste meritev, bodisi obravnavamo samo dolžine, samo kote ali oboje. V naši modificirani metodi tvorimo vse možne trikotnike v mreži in tako dobimo bistveno več informacij o spremembah v njej. Ob ponesrečenem izboru trikotnikov lahko spregledamo pomembne spremembe. Naravno je, da v trikotniku izračunamo dolžine in kote, zato tudi v našem predlogu najprej testiramo spremembo dolžin, nato spremembo kotov in šele na koncu spremembo kinematičnih parametrov v trikotniku. Ker izračun vseh mogočih kombinacij računsko ni več težava, je smiselno, da zajamemo vse informacije o spremembah v mreži. 2.3.1 Testiranje dolžin v geodetski mreži Testiranje izvedemo s testiranjem sprememb datumsko neodvisnih dolžin, ki povezujejo posamezno točko z drugimi točkami. Zapišemo ničelno in alternativno hipotezo (Welsch, 1982): H0: E(dldDij) = 0 … dolžina med točkama Pi in Pj se med dvema terminskima izmerama ni spremenila; Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 153 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Ha: E(dldDij) ≠ 0 … dolžina med točkama Pi in Pj se je med dvema terminskima izmerama spremenila. Sestavimo testno statistiko za razliko dolžin med točkama Pi in Pj v dveh izmerah (Welsch, 1982): 1 2 2 2 , ij ij ij ij dD dD dD D D dl Q dl T n s − = ⋅ (9) kjer je nD = 1 število prostostnih stopenj (Welsch, 1982; Soldo in Ambrožič, 2018). Spremembo dolžine izračunamo z izrazom dldDij = Dij2 − Dij1, (10) kjer se element dldDij nanaša na dolžini 1 1 1 1 1 2 2ˆ ˆ ˆ ˆ( ) ( )ij j i j iD y y x x= − + − in 2 2 2 2 2 2 2ˆ ˆ ˆ ˆ( ) ( )ij j i j iD y y x x= − + −  , ki ju izračunamo iz izravnanih koordinat med točkama Pi in Pj v predhodni in tekoči terminski izmeri. Element matrike kofaktorjev spremembe vrednosti dolžine v enačbi (9) izračunamo z enačbo (8) Qd Dij = Ld DijQud Dij L T d Dij , kjer vektor parcialnih odvodov meritev LdDij zapišemo kot (Welsch in Zhang, 1983; Soldo in Ambrožič, 2018): sin cos sin cos , ˆ ij ij dD dD ij ij ij ijν ν ν ν ∂   = = − −    ∂   l L x (11) kjer je νij = (νij1 + νij2) ⁄ 2 srednja vrednost smernih kotov νij1 in νij2, ki ju izračunamo iz izravnanih koordinat med točkama Pi in Pj, 1 1 2 2 1 2 1 1 2 2 ˆ ˆ ˆ ˆ arctan in arctan ˆ ˆ ˆ ˆ j i j i ij ij j i j i y y y y x x x x ν ν − − = = − − v predhodni in tekoči terminski izmeri. V podmatriki QudDij so pripadajoči elementi matrike Qu, ki se nanašajo na točki Pi in Pj. Testna statistika (9) se porazdeljuje po Fisherjevi porazdelitvi F1,f: – T 2 2Dij ≤ F1,f : ničelne hipoteze H0 ne moremo zavrniti in s tveganjem α ne moremo trditi, da se je dolžina med obravnavanima točkama spremenila, – T 2 2Dij > F1,f : ničelno hipotezo H0 zavrnemo in s tveganjem, manjšim od α, trdimo, da se je dolžina med obravnavanima točkama spremenila. 2.3.2 Testiranje kotov v geodetski mreži Testiranje izvedemo s testiranjem sprememb datumsko neodvisnih kotov, ki povezujejo posamezno točko z drugimi točkami. Zapišemo ničelno in alternativno hipotezo: H0: E(dldαijk) = 0 … kot med točkami Pi, Pj in Pk se med dvema terminskima izmerama ni spremenil; Ha: E(dldαijk) ≠ 0 … kot med točkami Pi, Pj in Pk se je med dvema terminskima izmerama spremenil. Sestavimo testno statistiko za kot med točkami Pi, Pj in Pk: 1 2 2 2 , ijk ijk ijk ijk d d ddl Q dl T n s α α α α α − = ⋅ (12) Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 154 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI| EN kjer je nα = 1 število prostostnih stopenj. Spremembo kota izračunamo z izrazom dldαijk = αijk2 − αijk1, (13) kjer se element dldαijk nanaša na kota αijk1 = νik1 − νij1 in αijk2 = νik2 − νij2 z vrhom v točki Pi proti točkam Pj in Pk v predhodni in tekoči terminski izmeri, pri čemer so νij1 in νik1, νij2 ter νik2 smerni koti iz izravnanih koordinat med točkama Pi in Pj ter Pi in Pk v predhodni in tekoči terminski izmeri. Element matrike kofaktorjev spremembe vrednosti kota v enačbi (12) izračunamo z enačbo (8) Qd αijk = Ld αijkQud αijk L T d αijk , kjer vektor parcialnih odvodov meritev Ldαijk zapišemo kot cos sin cos sincos sin cos sin , ˆ ijk ijk d ij ij ij ijik ik ik ik d ik ij ik ij ij ij ik ikD D D D D D D D α α ν ν ν νν ν ν ν ∂          = = − + + − −             ∂            l L x (14) kjer sta νij = (νij1 + νij2) ⁄ 2 in νik = (νik1 + νik2) ⁄ 2 srednji vrednosti smernih kotov νij1 in νik1, νij2 in νik2. Dolžini Dij = (Dij1 + Dij2) ⁄ 2 in Dik = (Dik1 + Dik2) ⁄ 2 predstavljata srednji vrednosti dolžin, ki ju izračunamo iz izravnanih koordinat med točkama Pi in Pj ter Pi in Pk v predhodni in tekoči terminski izmeri t1 in t2. V podmatriki Qudαijk so pripadajoči elementi matrike Qu, ki se nanašajo na točke Pi, Pj in Pk. Testna statistika (12) se porazdeljuje po Fisherjevi porazdelitvi F1,f: – T 2 2αijk ≤ F1,f : ničelne hipoteze H0 ne moremo zavrniti in s tveganjem α ne moremo trditi, da se je kot med obravnavanimi točkami spremenil, – T 2 2αijk > F1,f : ničelno hipotezo H0 zavrnemo in s tveganjem, manjšim od α, trdimo, da se je kot med obravnavanimi točkami spremenil. 2.3.3 Testiranje trikotnikov v geodetski mreži Obravnavano mrežo razdelimo na trikotnike in ugotavljamo, ali je posamezni trikotnik spremenil ob- liko med terminskima izmerama, s čimer pojasnimo koordinatne razlike v mreži. Zapišemo ničelno in alternativno hipotezo (Welsch, 1983): x x u 00 1 2ˆ ˆ: ( ) ( ) oz. ( )ijkH E E E ∆= = … koordinate točk v trikotniku se med dvema terminskima izmerama niso spremenile; x x u 01 2ˆ ˆ: ( ) ( ) oz. ( )ijkaH E E E ∆≠ ≠ … koordinate vsaj ene točke v trikotniku se so med dvema termin- skima izmerama spremenile. Sestavimo testno statistiko za oglišča Pi, Pj in Pk v trikotniku: 1 2 2 2 , ijk ijk ijk T n s ∆ ∆ − ∆ ∆ = ⋅ T uu Q u (15) kjer je n∆ = 3 število prostostnih stopenj (Welsch, 1983; Welsch in Zhang, 1983; Soldo in Ambrožič, 2018). Izračunamo vektor premikov oglišč v trikotniku ijk∆ u po enačbi (3), kjer sta x1ˆ in x 2ˆ vektorja izravnanih koordinat oglišč Pi, Pj in Pk v trikotniku po izravnavi predhodne in tekoče terminske izmere t1 in t2. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 155 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Matriko kofaktorjev koordinatnih razlik Qu izračunamo po enačbi (4), kjer sta x x x xQ Q1 1 2 2ˆ ˆ ˆ ˆin matriki kofaktorjev koordinatnih neznank oglišč Pi, Pj in Pk v trikotniku po izravnavi geodetske mreže. Testna statistika (15) se porazdeljuje po Fisherjevi porazdelitvi F1,f : – T 2 2∆ijk ≤ F3,f: ničelne hipoteze H0 ne moremo zavrniti in s tveganjem α ne moremo trditi, da so se koordinate v trikotniku z oglišči Pi, Pj in Pk spremenile, – T 2 2∆ijk > F3,f: ničelno hipotezo H0 zavrnemo in s tveganjem, manjšim od α, trdimo, da so se koordinate v trikotniku z oglišči Pi, Pj in Pk spremenile. V nadaljevanju lahko izračunamo kinematične parametre v posameznem trikotniku: p = Hu -1 . u, (16) kjer je Hu matrika deformacijskega modela, ki povezuje kinematične parametre s premiki točk v trikotniku uH ˆ ˆ ˆ0 1 0 ˆ ˆ ˆ0 0 1 ˆ ˆ ˆ0 1 0 ,ˆ ˆ ˆ0 0 1 ˆ ˆ ˆ0 1 0 ˆ ˆ ˆ0 0 1 i i i i i i j j j j j j k k k k k k x y y x y x x y y x y x x y y x y x −         − =       −      (17) PT ,xx xy yy x ye e e t tω =   (18) p pa je vektor kinematičnih parametrov, kjer so exx in eyy normalni deformaciji v smeri koordinatnih osi x in y, exy = eyx strižna deformacija (samo exx, eyy in exy so deformacijski parametri), ω predstavlja rotacijo, tx in ty pa translaciji v smeri koordinatnih osi x in y. Na podlagi osnovnih parametrov izračunamo še druge kinematične parametre (Welsch, 1983; Miha- ilović in Aleksić, 1994; Ašanin, 1986; Acar, 2010; Labant et al., 2014; Sušić et al., 2015b; Boresi in Sidebottom, 1985): ∆ = exx + eyy predstavlja spremembo ploščine, e1 = ½ (exx + eyy + ee) je glavna (največja) normalna deformacija, e2 = ½ (exx + eyy − ee) je glavna (najmanjša) normalna deformacija, kjer je ee2 = (exx − eyy) 2 + 4e2xy, 1 2 I 2 e e e − = je glavna strižna deformacija in γ = 2exy je inženirska strižna deformacija in predstavlja spremembo pravega kota med x in y smerjo. Smerna kota glavnih normalnih deformacij izračunamo z izrazom 2 tan 2 xy xx yy e e e ϑ = − , smerni kot glavne strižne deformacije pa z izrazom Ψ = ϑ + 45°. 3 RAČUNSKI PRIMER Zaradi enostavne in neposredne primerjave obravnavane modificirane metode deformacijske analize po postopku München z drugimi metodami deformacijske analize izberemo primer simulirane geodetske Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 156 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI| EN mreže, ki smo ga obravnavali z drugimi metodami deformacijske analize (Ambrožič, 2001; Ambrožič, 2004; Marjetič et al., 2012; Vrečko in Ambrožič, 2013; Soldo in Ambrožič, 2018; Ambrožič et al., 2019; Hamza et al., 2020; Batilović et al., 2022). V navedeni literaturi si lahko ogledamo skico mreže in dobimo vse potrebne podatke za izravnavo. Če na kratko povzamemo rezultate izravnave proste mre- že z redukcijo orientacijskih neznank predhodne izmere, dobimo s1 = 0,9699 in f1 = 30, tekoče izmere pa s2 = 1,1562 in f2 = 30. Ker ne moremo zavrniti identičnosti geodetskega datuma in homogenosti natančnosti v obravnavanih terminskih izmerah, izračunamo skupno referenčno varianco a posteriori s2 = 1,1387 in skupno število nadštevilnih meritev f = 60 (enačba 1). Povejmo še, da pri vseh uporabljenih testih v nadaljevanju izberemo tveganje α = 5 %. Testiranje skladnosti geodetske mreže je po klasičnem pristopu in po predlagani modificirani metodi deformacijske analize popolnoma enako, zato zapišimo le, da je T1 2 = 141,29 (enačba 2), kar presega kritično vrednost pri izbrani stopnji značilnosti testa F11,60 = 1,95. Tako zavrnemo ničelno hipotezo in s tveganjem, manjšim od α = 5 %, trdimo, da so se v mreži pojavile deformacije, in deformacijsko analizo nadaljujemo s testiranjem preoblikovanja geodetske mreže. Deformacijsko analizo po predlagani modificirani metodi nadaljujemo s testiranjem preoblikovanja geo- detske mreže, tako da najprej obravnavamo vse dolžine v geodetski mreži, teh je 21. Za vsako izračunano dolžino sestavimo testno statistiko T2 2 Dij (enačba 9) in jo primerjamo s kritično vrednostjo pri izbrani stopnji značilnosti testa F1,60 = 4,00, rezultate podajamo v preglednici 1. Izračunamo dejansko tveganje αT za zavrnitev ničelne hipoteze, ki ga primerjamo z izbranim tveganjem α = 5 %. Kjer je dejansko tveganje manjše od izbranega, moramo ničelno hipotezo zavrniti in trdimo, da so pomiki statistično značilni. V nasprotnih primerih pa lahko zaključimo, da se dolžine med mirujočimi točkami niso statistično značilno spremenile, saj je dejansko tveganje večje od 5 %. Preglednica 1: Analiza testiranja vseh dolžin med točkami Dolžina Med točkama T 22Dij H0 αT [%] Dolžina Med točkama T 22Dij H0 αT [%] 1 1-2 19.89 zavr. 0.00* 12 3-4 124.81 zavr. 0.00* 2 1-3 49.38 zavr. 0.00* 13 3-5 62.37 zavr. 0.00* 3 1-4 87.04 zavr. 0.00* 14 3-6 10.41 zavr. 0.20 4 1-5 64.84 zavr. 0.00* 15 3-7 9.20 zavr. 0.36 5 1-6 10.05 zavr. 0.24 16 4-5 0.08 ne z. 77.85 6 1-7 689.26 zavr. 0.00* 17 4-6 0.01 ne z. 91.54 7 2-3 109.75 zavr. 0.00* 18 4-7 186.39 zavr. 0.00* 8 2-4 84.02 zavr. 0.00* 19 5-6 0.63 ne z. 42.87 9 2-5 163.39 zavr. 0.00* 20 5-7 83.12 zavr. 0.00* 10 2-6 113.96 zavr. 0.00* 21 6-7 77.68 zavr. 0.00* 11 2-7 113.61 zavr. 0.00* * Dejansko tveganje je manjše od 0,005 %. Iz preglednice 1 in slike 1 vidimo, da se le dolžine med točkami 4-5, 4-6 in 5-6 niso statistično značilno spremenile. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 157 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Nato nadaljujemo deformacijsko analizo po predlagani modificirani metodi s testiranjem preoblikovanja geodetske mreže, tako da obravnavamo vse kote v geodetski mreži, teh je 105. Za vsak izračunan kot sestavimo testno statistiko T2 2 αijk (enačba 12) in jo primerjamo s kritično vrednostjo pri izbrani stopnji značilnosti testa F1,60 = 4,00, rezultate podajamo v preglednici 2. Izračunamo dejansko tveganje αT za za- vrnitev ničelne hipoteze in ga primerjamo z izbranim dopustnim tveganjem α = 5 %. Pri testiranju kotov ni mogoče enolično ugotoviti, kateri kot ne spremeni svoje vrednosti, saj testiramo tudi kote med dvema mirujočima in eno premično točko ali celo med eno mirujočo in dvema premičnima točkama. Iz pregle- dnice 2 lahko zaključimo, da je dejansko tveganje večje od 5 % pri vseh kombinacijah kotov, kjer nastopajo vse tri mirujoče točke, kot na primer v kotu 4-5-6 (αT = 33 %), 5-4-6 (αT = 44 %) in 6-4-5 (αT = 74 %). Slika 1: Analiza dolžin med točkami: z rdečo barvo so označene dolžine, ko ničelne hipoteze ne moremo zavrniti, s črno pa dolžine, ko ničelno hipotezo zavrnemo. Iz preglednice 2 in slike 2 vidimo, da se kar nekaj kotov med točkami ni statistično značilno spremenilo. Slika 2: Analiza kotov med točkami: z rdečo barvo so označeni koti, ko ničelne hipoteze ne moremo zavrniti, s črno pa koti, ko ničelno hipotezo zavrnemo. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 158 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI| EN Preglednica 2: Analiza testiranja vseh kotov med točkami Kot Med točkami T 22αijk H0 αT [%] Kot Med točkami T 22αijk H0 αT [%] 1 1-2-3 765.25 zavr. 0.00* 36 3-2-4 172.51 zavr. 0.00* 2 1-2-4 382.51 zavr. 0.00* 37 3-2-5 256.93 zavr. 0.00* 3 1-2-5 377.49 zavr. 0.00* 38 3-2-6 322.97 zavr. 0.00* 4 1-2-6 409.63 zavr. 0.00* 39 3-2-7 15.28 zavr. 0.02 5 1-2-7 304.97 zavr. 0.00* 40 3-4-5 2.50 ne z. 11.88 6 1-3-4 55.84 zavr. 0.00* 41 3-4-6 4.70 zavr. 3.42 7 1-3-5 2.08 ne z. 15.41 42 3-4-7 183.15 zavr. 0.00* 8 1-3-6 38.50 zavr. 0.00* 43 3-5-6 2.24 ne z. 14.01 9 1-3-7 32.56 zavr. 0.00* 44 3-5-7 368.79 zavr. 0.00* 10 1-4-5 24.42 zavr. 0.00* 45 3-6-7 342.61 zavr. 0.00* 11 1-4-6 104.38 zavr. 0.00* 46 4-1-2 153.77 zavr. 0.00* 12 1-4-7 0.33 ne z. 56.77 47 4-1-3 7.99 zavr. 0.64 13 1-5-6 78.35 zavr. 0.00* 48 4-1-5 0.05 ne z. 83.07 14 1-5-7 20.96 zavr. 0.00* 49 4-1-6 3.88 ne z. 5.36 15 1-6-7 101.10 zavr. 0.00* 50 4-1-7 33.76 zavr. 0.00* 16 2-1-3 844.94 zavr. 0.00* 51 4-2-3 17.86 zavr. 0.01 17 2-1-4 474.80 zavr. 0.00* 52 4-2-5 42.39 zavr. 0.00* 18 2-1-5 448.55 zavr. 0.00* 53 4-2-6 69.33 zavr. 0.00* 19 2-1-6 406.03 zavr. 0.00* 54 4-2-7 22.08 zavr. 0.00* 20 2-1-7 916.07 zavr. 0.00* 55 4-3-5 4.18 zavr. 4.52 21 2-3-4 361.72 zavr. 0.00* 56 4-3-6 3.20 ne z. 7.89 22 2-3-5 426.83 zavr. 0.00* 57 4-3-7 0.28 ne z. 59.82 23 2-3-6 456.29 zavr. 0.00* 58 4-5-6 0.98 ne z. 32.59 24 2-3-7 76.78 zavr. 0.00* 59 4-5-7 14.57 zavr. 0.03 25 2-4-5 69.08 zavr. 0.00* 60 4-6-7 15.39 zavr. 0.02 26 2-4-6 132.49 zavr. 0.00* 61 5-1-2 1.17 ne z. 28.29 27 2-4-7 37.28 zavr. 0.00* 62 5-1-3 4.23 zavr. 4.40 28 2-5-6 68.82 zavr. 0.00* 63 5-1-4 5.51 zavr. 2.23 29 2-5-7 230.59 zavr. 0.00* 64 5-1-6 5.41 zavr. 2.34 30 2-6-7 365.41 zavr. 0.00* 65 5-1-7 230.42 zavr. 0.00* 31 3-1-2 561.33 zavr. 0.00* 66 5-2-3 15.02 zavr. 0.03 32 3-1-4 0.39 ne z. 53.44 67 5-2-4 4.47 zavr. 3.86 33 3-1-5 1.06 ne z. 30.63 68 5-2-6 1.15 ne z. 28.79 34 3-1-6 11.15 zavr. 0.14 69 5-2-7 232.58 zavr. 0.00* 35 3-1-7 468.25 zavr. 0.00* 70 5-3-4 38.75 zavr. 0.00* Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 159 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Kot Med točkami T 22αijk H0 αT [%] Kot Med točkami T 22αijk H0 αT [%] 71 5-3-6 9.61 zavr. 0.30 89 6-4-7 127.84 zavr. 0.00* 72 5-3-7 240.57 zavr. 0.00* 90 6-5-7 90.46 zavr. 0.00* 73 5-4-6 0.60 ne z. 44.04 91 7-1-2 132.04 zavr. 0.00* 74 5-4-7 73.79 zavr. 0.00* 92 7-1-3 199.15 zavr. 0.00* 75 5-6-7 106.42 zavr. 0.00* 93 7-1-4 9.90 zavr. 0.26 76 6-1-2 296.34 zavr. 0.00* 94 7-1-5 41.02 zavr. 0.00* 77 6-1-3 35.57 zavr. 0.00* 95 7-1-6 33.70 zavr. 0.00* 78 6-1-4 98.75 zavr. 0.00* 96 7-2-3 27.83 zavr. 0.00* 79 6-1-5 69.53 zavr. 0.00* 97 7-2-4 44.72 zavr. 0.00* 80 6-1-7 314.80 zavr. 0.00* 98 7-2-5 233.68 zavr. 0.00* 81 6-2-3 258.54 zavr. 0.00* 99 7-2-6 207.06 zavr. 0.00* 82 6-2-4 38.20 zavr. 0.00* 100 7-3-4 215.43 zavr. 0.00* 83 6-2-5 22.19 zavr. 0.00* 101 7-3-5 432.37 zavr. 0.00* 84 6-2-7 20.63 zavr. 0.00* 102 7-3-6 332.44 zavr. 0.00* 85 6-3-4 97.78 zavr. 0.00* 103 7-4-5 129.96 zavr. 0.00* 86 6-3-5 31.15 zavr. 0.00* 104 7-4-6 57.94 zavr. 0.00* 87 6-3-7 321.56 zavr. 0.00* 105 7-5-6 5.96 zavr. 1.76 88 6-4-5 0.11 ne z. 74.42 * Dejansko tveganje je manjše od 0,005 %. Deformacijsko analizo nadaljujemo po predlagani modificirani metodi še s testiranjem preoblikovanja geodetske mreže, tako da obravnavamo skladnost vseh trikotnikov v geodetski mreži, teh je 35. Za vsak izračunan trikotnik sestavimo testno statistiko T2 2 ∆ijk (enačba 15) in jo primerjamo s kritično vrednostjo pri izbrani stopnji značilnosti testa F3,60 = 2,76, rezultate podajamo v preglednici 3. Izračunamo dejansko tveganje αT za zavrnitev ničelne hipoteze, ki ga primerjamo z izbranim dopustnim tveganjem α = 5 %. Ugotovimo lahko, da se trikotnik med mirujočimi točkami 4-5-6 ni statistično značilno spremenil, saj je dejansko tveganje bistveno večje od 5 % in znaša αT = 77 %. Preglednica 3: Analiza testiranja skladnosti trikotnikov med točkami Trikotnik Med točkami T 22∆ijk H0 αT [%] Trikotnik Med točkami T 22∆ijk H0 αT [%] 1 1-2-3 286.17 zavr. 0.00* 10 1-4-5 35.00 zavr. 0.00* 2 1-2-4 159.09 zavr. 0.00* 11 1-4-6 46.79 zavr. 0.00* 3 1-2-5 157.59 zavr. 0.00* 12 1-4-7 271.23 zavr. 0.00* 4 1-2-6 166.60 zavr. 0.00* 13 1-5-6 33.96 zavr. 0.00* 5 1-2-7 336.48 zavr. 0.00* 14 1-5-7 249.07 zavr. 0.00* 6 1-3-4 62.88 zavr. 0.00* 15 1-6-7 229.98 zavr. 0.00* 7 1-3-5 42.33 zavr. 0.00* 16 2-3-4 136.82 zavr. 0.00* 8 1-3-6 24.16 zavr. 0.00* 17 2-3-5 158.79 zavr. 0.00* 9 1-3-7 278.24 zavr. 0.00* 18 2-3-6 163.70 zavr. 0.00* Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 160 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI| EN Trikotnik Med točkami T 22∆ijk H0 αT [%] Trikotnik Med točkami T 22∆ijk H0 αT [%] 19 2-3-7 62.03 zavr. 0.00* 28 3-4-7 116.79 zavr. 0.00* 20 2-4-5 60.86 zavr. 0.00* 29 3-5-6 20.86 zavr. 0.00* 21 2-4-6 68.60 zavr. 0.00* 30 3-5-7 162.34 zavr. 0.00* 22 2-4-7 110.69 zavr. 0.00* 31 3-6-7 140.22 zavr. 0.00* 23 2-5-6 67.44 zavr. 0.00* 32 4-5-6 0.37 ne z. 77.30 24 2-5-7 147.62 zavr. 0.00* 33 4-5-7 95.96 zavr. 0.00* 25 2-6-7 133.31 zavr. 0.00* 34 4-6-7 98.21 zavr. 0.00* 26 3-4-5 42.87 zavr. 0.00* 35 5-6-7 56.68 zavr. 0.00* 27 3-4-6 41.92 zavr. 0.00* * Dejansko tveganje je manjše od 0,005 %. Slike trikotnikov med točkami ne prikažemo, saj je zaradi velikega števila trikotnikov nepregledna. Zaradi testiranja velikega števila vseh dolžin med točkama v mreži, še večjega števila vseh kotov med točkami v mreži in tudi velikega števila testiranja skladnosti vseh mogočih trikotnikov v mreži, kar je bistvo predlagane modificirane metode deformacijske analize po postopku München, je analiza težavna in nepregledna. Lahko pa jo poenostavimo in njeno preglednost povečamo tako, da vsa tri testiranja količin naredimo na enoto, to je na posamezen trikotnik. V preglednici 4 prikazujemo rezultate testiranja vseh treh količin v posameznem trikotniku. Preglednica 4: Analiza testiranja vseh treh količin v trikotnikih Trikotnik Med točkami i-j-k H0-D i-j j-k i-k H0-α i- j-k j-i-k k-i-j H0-∆ i-j-k 1 1-2-3 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 2 1-2-4 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 3 1-2-5 zavr. zavr. zavr. zavr. zavr. ne z. zavr. 4 1-2-6 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 5 1-2-7 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 6 1-3-4 zavr. zavr. zavr. zavr. ne z. zavr. zavr. 7 1-3-5 zavr. zavr. zavr. ne z. ne z. zavr. zavr. 8 1-3-6 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 9 1-3-7 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 10 1-4-5 zavr. ne z. zavr. zavr. ne z. zavr. zavr. 11 1-4-6 zavr. ne z. zavr. zavr. ne z. zavr. zavr. 12 1-4-7 zavr. zavr. zavr. ne z. zavr. zavr. zavr. 13 1-5-6 zavr. ne z. zavr. zavr. zavr. zavr. zavr. 14 1-5-7 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 15 1-6-7 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 16 2-3-4 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 17 2-3-5 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 18 2-3-6 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 19 2-3-7 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 20 2-4-5 zavr. ne z. zavr. zavr. zavr. zavr. zavr. 21 2-4-6 zavr. ne z. zavr. zavr. zavr. zavr. zavr. 22 2-4-7 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 23 2-5-6 zavr. ne z. zavr. zavr. ne z. zavr. zavr. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 161 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N Trikotnik Med točkami i-j-k H0-D i-j j-k i-k H0-α i- j-k j-i-k k-i-j H0-∆ i-j-k 24 2-5-7 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 25 2-6-7 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 26 3-4-5 zavr. ne z. zavr. ne z. zavr. zavr. zavr. 27 3-4-6 zavr. ne z. zavr. zavr. ne z. zavr. zavr. 28 3-4-7 zavr. zavr. zavr. zavr. ne z. zavr. zavr. 29 3-5-6 zavr. ne z. zavr. ne z. zavr. zavr. zavr. 30 3-5-7 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 31 3-6-7 zavr. zavr. zavr. zavr. zavr. zavr. zavr. 32 4-5-6 ne z. ne z. ne z. ne z. ne z. ne z. ne z. 33 4-5-7 ne z. zavr. zavr. zavr. zavr. zavr. zavr. 34 4-6-7 ne z. zavr. zavr. zavr. zavr. zavr. zavr. 35 5-6-7 ne z. zavr. zavr. zavr. zavr. zavr. zavr. Slika 3: Izris premikov točk: z rdečo barvo so označeni premiki na točkah 4, 5 in 6, ko ničelne hipoteze ne moremo zavrniti, s črno pa premiki, ko ničelno hipotezo zavrnemo. Na sliki 3 prikazujemo razlike izravnanih koordinat obeh terminskih izmer in ne deformacijskih para- metrov za posamezni trikotnik, kar je eden od rezultatov deformacijske analize po postopku München (Welsch in Zhang, 1983; Welsch, 1983; Soldo in Ambrožič, 2018; Sušić et al., 2015b; Sušić et al., 2017; Mihailović in Aleksić, 1994). Trikotnikov je preveč, nekaj jih je zelo iztegnjenih, zato bi bila slika zelo nepregledna. Na podlagi tako zapisanih rezultatov testiranja v preglednici 4 pa hitro ugotovimo, da se vse dolžine, vsi koti in skladnost v trikotniku 32 niso statistično značilno spremenili, kar z izrisom premikov prikažemo na sliki 3. Z izbranim dopustnim tveganjem α = 5 % trdimo, da se tudi točke tega trikotnika niso stati- stično značilno premaknile, saj smo trikotnik testirali tako glede spremembe dolžin in kotov v trikotniku kakor tudi skladnosti celotnega trikotnika. Po enačbi (18) izračunamo osnovne kinematične parametre in jih prikažemo v preglednici 5. Drugih deformacijskih parametrov ne izračunamo in prikazujemo, saj si jih lahko po zgoraj zapisanih enačbah enostavno izračunamo. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 162 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI| EN Preglednica 5: Izračunani kinematični parametri v vseh trikotnikih Trikotnik Med točkami i -j-k exx [μs] exy [μs] eyy [μs] ω [''] tx [mm] ty [mm] 1 1-2-3 -161.27 82.79 -18.70 -2.7 0.021 -0.062 2 1-2-4 -43.92 57.70 -18.70 -7.9 -0.096 -0.012 3 1-2-5 11.14 50.49 -18.70 -9.4 -0.151 0.003 4 1-2-6 151.73 50.35 -18.70 -9.4 -0.292 0.003 5 1-2-7 46.19 76.48 -18.70 -4.0 -0.186 -0.049 6 1-3-4 52.09 -22.84 32.64 0.3 -0.072 -0.022 7 1-3-5 24.18 -4.11 20.39 0.9 -0.060 -0.031 8 1-3-6 39.05 5.69 4.66 4.6 -0.066 -0.044 9 1-3-7 158.66 -16.92 -7.75 13.8 -0.119 -0.053 10 1-4-5 21.33 8.38 2.64 -1.2 -0.080 -0.016 11 1-4-6 43.04 0.08 -10.54 2.7 -0.074 -0.013 12 1-4-7 368.00 -113.88 -233.38 102.9 0.007 0.031 13 1-5-6 26.76 -12.01 -18.44 3.5 -0.042 0.003 14 1-5-7 2.38 79.05 85.25 17.9 -0.212 -0.088 15 1-6-7 86.78 33.95 13.46 0.6 -0.162 -0.061 16 2-3-4 -14.58 -58.59 75.42 13.5 0.315 -0.187 17 2-3-5 -62.75 -9.55 36.67 8.7 0.218 -0.136 18 2-3-6 -98.67 29.36 0.77 5.6 0.146 -0.088 19 2-3-7 -54.19 -0.77 -8.92 13.2 0.235 -0.075 20 2-4-5 -32.33 12.71 4.05 0.8 0.066 -0.054 21 2-4-6 -34.60 22.41 -13.45 -0.7 0.034 -0.021 22 2-4-7 -28.08 1.59 -68.24 5.1 0.126 0.081 23 2-5-6 -21.31 18.08 -18.57 -2.7 0.011 0.002 24 2-5-7 -129.06 -193.66 -434.50 -1.9 0.550 1.042 25 2-6-7 13.88 58.63 6.90 3.0 -0.050 -0.116 26 3-4-5 81.75 -12.17 -2.07 -6.7 -0.244 0.112 27 3-4-6 55.49 -26.77 13.21 -1.6 -0.092 0.053 28 3-4-7 20.50 -57.18 0.71 3.0 0.111 0.102 29 3-5-6 14.21 -15.96 11.03 -0.5 -0.028 0.029 30 3-5-7 -83.15 -50.90 0.43 2.3 0.284 0.096 31 3-6-7 1279.93 35.50 -26.54 45.7 -1.984 -0.397 32 4-5-6 -5.80 0.43 1.32 -2.3 -0.006 0.020 33 4-5-7 -57.45 -17.02 -1.65 -4.8 0.135 0.101 34 4-6-7 -119.44 -3.87 34.03 14.9 0.153 0.112 35 5-6-7 -39.65 11.85 25.68 -8.7 0.031 0.041 4 SKLEP V preteklih raziskavah smo dokazali, da so rezultati zelo podobni ne glede na to, katero metodo de- formacijske analize uporabimo na testnem primeru (Ambrožič, 2001; Ambrožič, 2004; Marjetič et al., 2012; Vrečko in Ambrožič, 2013; Soldo in Ambrožič, 2018; Ambrožič et al., 2019; Hamza et al., 2020; Batilović et al., 2022). V naši raziskavi smo se osredotočili na pomembne izboljšave, ki bi omogočile Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 163 | GEODETSKI VESTNIK | 67/2 | RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI | E N zanesljivejše ugotavljanje deformacij obravnavanega objekta oziroma merjenega območja. V modificirani metodi deformacijske analize po postopku München pred fazo preoblikovanja geodetske mreže, torej pred tvorjenjem trikotnikov v mreži, predlagamo, da poleg testiranja sprememb neodvisnih dolžin izvedemo tudi testiranje neodvisnih kotov med vsemi točkami v mreži. V naslednji fazi pa tvorimo vse možne trikotnike med točkami v mreži in ne le izbranih, saj tako pridobimo zanesljivejšo informacijo o more- bitnih spremembah oblike trikotnikov in posledično o točkah, ki so se statistično značilno premaknile. Pri analizi testiranja vseh dolžin v mreži nam je s predlagano metodo uspelo najti dolžine med točkami 4-5, 4-6 in 5-6, ki se niso statistično značilno spremenile, in zato ničelne hipoteze ne moremo zavrniti (preglednica 1 in slika 1). To je skladno s simuliranimi premiki v mreži, ko so točke 4, 5 in 6 obravnavane kot mirujoče. Pri analizi testiranja vseh kotov v mreži nam je s predlagano metodo uspelo najti vse kombinacije kotov med točkami 4-5-6, 5-4-6 in 6-4-5, ki se niso statistično značilno spremenili, in zato ničelne hipoteze ne moremo zavrniti (preglednica 2 in slika 2). Pri analizi testiranja skladnosti vseh trikotnikov v mreži nam je s predlagano metodo uspelo najti trikotnik z oglišči v točkah 4, 5 in 6, ki se ni statistično značilno spremenil, in zato ničelne hipoteze ne moremo zavrniti (preglednica 3). Če rezultate testiranj zapišemo bolj pregledno, lahko ugotovimo, da se vse dolžine, vsi koti in skladnost v trikotniku 32, torej med točkami 4-5-6, niso statistično značilno spremenili (preglednica 4). Sklepamo, da z dopustnim tveganjem α = 5 % ne moremo trditi, da so se točke tega trikotnika premaknile, kar je skladno s simuliranimi premiki v mreži. Prednosti predlagane modificirane metode so, da uporabnik deformacijske analize navadno tvori: – trikotnike med sosednjimi točkami, mi pa predlagamo tvorjenje trikotnikov med vsemi točkami, s čimer dobimo informacijo o deformacijah med vsemi točkami, ki jo lahko spregledamo, če tri- kotnike tvorimo sami; – »lepe« oblike trikotnikov, kar spet vodi do tega, da v zelo iztegnjenem trikotniku ali v trikotniku, kjer je ena stranica veliko krajša od drugih dveh, izgubimo informacijo o deformacijah takega trikotnika. Slabost predlagane modificirane metode se pokaže pri geodetski mreži, ko mirujeta samo dve točki. V tem primeru ne moremo tvoriti trikotnika (s tremi točkami), ko se vse tri testirane količine (dolžine, koti in skladnost trikotnika) ne bi statistično značilno spremenile. Glede na predlagano metodo lahko ugotovimo, da je mirujočih točk premalo. Ocenjujemo, da predlagana modificirana metoda vključuje pomembne izboljšave, ki nam pomagajo bolj zanesljivo najti stabilne točke oziroma statistično značilne deformacije obravnavanega objekta, kar je bistvo vsake deformacijske analize. Glede na napredne možnosti računskih operacij in shranjevanja rezultatov izračunov sklepamo, da prostorska in časovna zahtevnost predlaganega algoritma ni problematična. Zahvala Prispevek je nastal v okviru raziskovalnega programa P2-0227: Geoinformacijska infrastruktura in trajno- stni prostorski razvoj Slovenije in temeljnega raziskovalnega projekta J2-2489: SLOKIN – Geokinematski model ozemlja Slovenije, ki ju financira Javna agencija za raziskovalno dejavnost Republike Slovenije. Literatura in viri Glej stran 147. Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 | | 164 | | 67/2 | GEODETSKI VESTNIK RE CE NZ IRA NI ČL AN KI | P EE R- RE VIE W ED AR TIC LE S SI| EN doc. dr. Simona Savšek Univerza v Ljubljani, Fakulteta za gradbeništvo in geodezijo Jamova cesta 2 1000 Ljubljana e-naslov: simona.savsek@fgg.uni-lj.si izr. prof. dr. Tomaž Ambrožič Univerza v Ljubljani, Fakulteta za gradbeništvo in geodezijo Jamova cesta 2 1000 Ljubljana e-naslov: tomaz.ambrozic@fgg.uni-lj.si Savšek S., Ambrožič T. (2023). Modificirana metoda deformacijske analize po postopku München. Geodetski vestnik, 67 (2), 131-164. DOI: https://doi.org/10.15292/geodetski-vestnik.2023.02.131-164 Simona Savšek, Tomaž Ambrožič | MODIFICIRANA METODA DEFORMACIJSKE ANALIZE PO POSTOPKU MÜNCHEN | MODIFIED METHOD OF DEFORMATION ANALYSIS ACCORDING TO THE MUNICH APPROACH | 131-164 |